Percolation in interdependent and interconnected networks: Abrupt change from second- to first-order transitions

Robustness of two coupled networks systems has been studied separately only for dependency coupling Buldyrev et al., Nature (London) 464, 1025 (2010) and only for connectivity coupling Leicht and D'Souza, e-print arXiv:0907.0894. Here we study, using a percolation approach, a more realistic coupled networks system where both interdependent and interconnected links exist. We find rich and unusual phase-transition phenomena including hybrid transition of mixed first and second order, i.e., discontinuities like in a first-order transition of the giant component followed by a continuous decrease to zero like in a second-order transition. Moreover, we find unusual discontinuous changes from second-order to first-order transition as a function of the dependency coupling between the two networks.